Genus of the Cartesian Product of Triangles

We investigate the orientable genus of Gn, the cartesian product of n triangles, with a particular attention paid to the two smallest unsolved cases n = 4 and 5. Using a lifting method we present a general construction of a low-genus embedding of Gn using a low-genus embedding of Gn−1. Combining this method with a computer search and a careful analysis of face structure we show that 30 6 γ(G4) 6 37 and 133 6 γ(G5) 6 190. Moreover, our computer search resulted in more than 1300 nonisomorphic minimum-genus embeddings of G3. We also introduce genus range of a group and (strong) symmetric genus range of a Cayley graph and of a group. The (strong) symmetric genus range of irredundant Cayley graphs of Zn p is calculated for all odd primes p.